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Oscillation of linear Hamiltonian systems. (English) Zbl 1047.34030

The authors investigate oscillatory properties of the linear Hamiltonian system

x ' =A(t)x+B(t)u,u ' =C(t)x-A T (t)u,(*)

where A,B,C are n×n-matrices with continuous entries for t[t 0 ,), B,C are symmetric and the matrix B is supposed to be positive definite. Using a transformation which preserves the oscillatory nature of transformed systems, system (*) is transformed into the “diagonal off” system y ' =B ˜(t)z, z ' =-C ˜(t)y with B ˜ positive definite, hence this system is equivalent to the second-order vector-matrix Sturm-Liouville differential equation

(B ˜ -1 (t)y ' ) ' +C ˜(t)y=0·(**)

The main result of the paper is formulated under the assumption that the matrix B ˜(t)-I is nonnegative definite (I denotes the identity matrix). Under this assumption, system (**) is a Sturmian majorant of the system y '' +C ˜(t)y=0. So, oscillation of the last system implies oscillation of (**) and hence, in turn, also of (*). From this point of view, the results of the paper are very close to those presented in the paper of G. J. Butler, L. H. Erbe and A. B. Mingarelli [Trans. Am. Math. Soc. 303, 263–282 (1987; Zbl 0648.34031)].

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A30Linear ODE and systems, general